Does a spherical triangle with 2 equal sides necessarily have 2 base angles of size $\pi/2$? The reason I think this is that if we have a triangle $ABC$ and $AB=AC$ (in spherical distance), we could view it as $A$ being at the center of a spherical disk with radius $|AB|$ then the given description would be realised by any triangle we draw by drawing 2 spherical straight lines from $A$ and intersecting the circumference of the disk. But spherical straight lines would intersect the circumference at $\pi/2$ like how the altitudes intersect the latitudes on a globe, right?
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Your last sentence is correct; the lines would intersect the circumference at right angles. However, your further conclusion is incorrect, since the circumference typically isn't a great circle, so the line joining $B$ and $C$ isn't the circumference.
An easy counter-example is to look at an equilateral triangle which has two equal sides and angles (three actually, but that is a bonus).
The angles of a spherical triangle sum to something between $\pi$ and $5\pi$, so the angles of an equilateral triangle can be anything between $\pi/3$ and $5\pi/3$. They do not have to be $\pi/2$; if they are then you are looking at an octant of the sphere.